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Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions. A preliminary integration by parts may be necessary. f(t) = tcos2t

Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions. A preliminary integration by parts may be necessary. f(t) = tcos2t

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions. A preliminary integration by parts may be necessary.

f(t) = tcos2t

I have this hint: Use the formula L{sin⁡kt }=k/(s^2+k^2 ), and then differentiate both sides over k(!), but I don't understand how to use it.

Please break this down for me, using the hint, if you get it, or not, otherwise, and please do not skip steps, assuming that I know what you did when you skipped them, if you know what I mean.

Thank you for your help.

1 f(t) t" (n ≥ 0) ta (a > -1) eat cos kt sin kt cosh kt sinh kt u(ta) 1 S 1 5² n! sn+1 Γ(α + 1) sa+1 1 sa F(s) S s²+k² k s²+k² S s²k² k s² - k² e-as S (s > 0) (s > 0) (s > 0) (s > 0) (s > 0) (s > 0) (s > 0) (s > |k|) (s > |k|) (s > 0) FIGURE 4.1.2. A short table of Laplace transforms.
Transcribed Image Text:1 f(t) t" (n ≥ 0) ta (a > -1) eat cos kt sin kt cosh kt sinh kt u(ta) 1 S 1 5² n! sn+1 Γ(α + 1) sa+1 1 sa F(s) S s²+k² k s²+k² S s²k² k s² - k² e-as S (s > 0) (s > 0) (s > 0) (s > 0) (s > 0) (s > 0) (s > 0) (s > |k|) (s > |k|) (s > 0) FIGURE 4.1.2. A short table of Laplace transforms.
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